binomial expansion of (1+xsq)raise to power-1 at x=1

Question

Binomial expansion of (1+xsq) with power-1 at x=1

Answer 1

Binomial expansion:

(1+x²)^-1=1+(-1)x²+(-1)(-2)x⁴/2!+(-1)(-2)(-3)x⁶/3!+...+(-1)ⁿn!x²ⁿ/n!=

1-x²+x⁴-x⁶+...

If x=1, we get 1-1+1-1+... which is either 0 or 1, so is indeterminate as a series. However, if x<1, the sum of this geometric progression is:

1/(1+x²), the original function. As x→1 the limit is ½. When x≥1 the sum is indeterminate or is infinite.